Bob Bradley's search for a first Swansea win goes on after Seamus Coleman's late header earned Everton a 1-1 draw at Goodison Park.
Swansea were heading for their first Premier League win since the opening day of the season, but Coleman rose highest to equalise in the 89th minute.
Gylfi Sigurdsson had given Swansea the lead from the spot in the first half after he was brought down by Phil Jagielka.
The draw leaves Swansea rock-bottom on six points, while Everton have won just one league game in eight attempts.
While Everton welcomed back Leighton Baines and James McCarthy to the starting line-up, Bradley made eight changes to his Swansea side that lost to Manchester United a fortnight ago.
And it was a decent start from the visitors, with a Sigurdsson free-kick from 20 yards out forcing a fine save from Maarten Stekelenburg.
Everton were slow to get going but then carved out two glorious opportunities to take the lead.
Firstly, a powerful Yannick Bolasie cross ricocheted off Romelu Lukaku's thigh in the six-yard box and sailed over the bar when it could have gone anywhere.
Moments later, Ross Barkley started and finished a fine Everton move, but shot wide on the turn following Bolasie's pinpoint cross.
Five minutes before the break, Barrow picked out Sigurdsson in the box, who flicked it past Jagielka before the Everton defender bundled the forward over as he was getting a shot away.
Jagielka received a yellow, while Sigurdsson picked himself up to take the spot-kick, sending Stekelenburg the wrong way with a fine penalty into the top corner.
After the break, Everton enjoyed more of the ball, but in truth, they created better chances in the first half.
Swansea did have the chance to go two goals clear just after the hour-mark, but Jay Fulton glanced his header wide from Sigurdsson's cross.
A resolute Swansea defence appeared to be guiding their side to victory, but Everton had the final say in the last minute of normal time.
Ashley Williams, who was facing his old club for the first time since a £12m summer move, sent in a cross which was cleared out to Bolasie.
The winger's cross was headed into the air by Jordi Amat, and Coleman leaped highest to direct the ball into the far corner beyond Fabianski's grasp.
Everton: Stekelenburg (6), Baines (6), Jagielka (6), Williams (6), Coleman (7), McCarthy (7), Gana (7), Bolasie (7), Lennon (6), Barkley (8), Lukaku (7).
Subs: Deulofeu (6), Mirallas (6), Valencia (N/A).
Swansea: Fabianski (7), Taylor (7), Amat (8), Fernandez (7), Naughton (7), Fer (6), Cork (7), Fulton (7), Routledge (7), Sigurdsson (7), Barrow (7).
Subs: Ki (N/A), Dyer (6).
Man of the match: Ross Barkley.